Networks of coupled degrade-and-fire (DF) oscillators are simple dynamical models of assemblies of interacting self-repressing genes. For mean-field interactions, which most mathematical studies have assumed so far, every trajectory must approach a periodic orbit. Moreover, asymptotic cluster distributions can be computed explicitly in terms of coupling intensity, and a massive collection of di...

A scheme of matched asymptotic expansions is used to obtain approximations to the dispersion relation when waves, governed by the Helmholtz equation, propagate through a two-dimensional lattice of scatterers on each of which a homogeneous Dirichlet boundary condition is imposed. The scatterers must be identical, but can be of any shape as long as each is small relative to the wavelength and the...

We give a complete picture regarding the asymptotic periodicity of positive solutions of the following difference equation: xn = f (xn−p1 , . . . ,xn−pk ,xn−q1 , . . . ,xn−qm), n∈N0, where pi, i ∈ {1, . . . ,k}, and qj , j ∈ {1, . . . ,m}, are natural numbers such that p1 < p2 < ··· < pk, q1 < q2 < ··· < qm and gcd(p1, . . . , pk,q1, . . . ,qm) = 1, the function f ∈ C[(0,∞), (α,∞)], α > 0, is i...

Assume that (AI) X is a real Banach space. (A2) X+ is a closed subset of X with the following properties: (i) if xe X+, y € X+, a e [0, oo) then x + y & X+ and ax € X+; (ii) there exists Mo € (0, oo) such that for each x € X there exist x+ e X+ and x_ 6 X+ which satisfy _=_+-__, ||_+|| < Mo||x||, ||x_||<Mo||x|| and if x = j/+ — j/_ for some y+ 6 X+, j/_ _ X+ then y+ — x.|6 X+; (iii) if x e X+, ...

We study the dynamics of discrete-time regulatory networks on random digraphs. For this we define ensembles of deterministic orbits of random regulatory networks, and introduce some statistical indicators related to the long-term dynamics of the system. We prove that, in a random regulatory network, initial conditions almost certainly converge to a periodic attractor. We study the subnetworks, ...